Collinearly-improved BK evolution meets the HERA data

E. Iancua,∗, J.D. Madrigala, A.H. Muellerb, G. Soyeza, D.N. Triantafyllopoulosc

aInstitut de Physique Theorique, CEA Saclay, CNRS UMR 3681, F-91191 Gif-sur-Yvette, FrancebDepartment of Physics, Columbia University, New York, NY 10027, USA

cEuropean Centre for Theoretical Studies in Nuclear Physics and Related Areas (ECT*)and Fondazione Bruno Kessler, Strada delle Tabarelle 286, I-38123 Villazzano (TN), Italy

Abstract

In a previous publication, we have established a collinearly-improved version of the Balitsky-Kovchegov(BK) equation, which resums to all orders the radiative corrections enhanced by large double transverselogarithms. Here, we study the relevance of this equation as a tool for phenomenology, by confrontingit to the HERA data. To that aim, we first improve the perturbative accuracy of our resummation, byincluding two classes of single-logarithmic corrections: those generated by the first non-singular terms in theDGLAP splitting functions and those expressing the one-loop running of the QCD coupling. The equationthus obtained includes all the next-to-leading order corrections to the BK equation which are enhanced by(single or double) collinear logarithms. We then use numerical solutions to this equation to fit the HERAdata for the electron-proton reduced cross-section at small Bjorken x. We obtain good quality fits forphysically acceptable initial conditions. Our best fit, which shows a good stability up to virtualities as largeas Q2 = 400 GeV2 for the exchanged photon, uses as an initial condition the running-coupling version of theMcLerran-Venugopalan model, with the QCD coupling running according to the smallest dipole prescription.

Keywords: QCD, High-energy evolution, Parton saturation, Deep Inelastic Scattering

1. Introduction

The wealth of data on electron-proton deep inelastic scattering collected by the experiments at HERAover 15 years of operation has allowed for stringent tests of our understanding of high-energy scatteringfrom first principles. This refers in particular to the ‘small-x’ regime where perturbative QCD predicts arapid growth of the gluon density with increasing energy (or decreasing Bjorken x), leading to non-linearphenomena like multiple scattering and gluon saturation [1, 2]. The simplicity of the dipole factorizationfor deep inelastic scattering at high energy [3, 4] has favored the emergence of relatively simple ‘dipolemodels’, in which the high-density effects are efficiently implemented as unitarity corrections to the cross-section for the scattering between a quark-antiquark dipole and the proton. Such models allowed for rathersuccessful fits to the small-x HERA data at a time where the theory of the non-linear evolution in QCD wasinsufficiently developed and the pertinence of gluon saturation for the phenomenology was far from beingwidely accepted. The first such model — the “GBW saturation model” [5, 6] — provided a rather gooddescription of the early HERA data for the inclusive and diffractive structure functions at x ≤ 10−2 withonly 3 free parameters. This success inspired new ways to look at the HERA data, which in particularled to the identification of geometric scaling [7]. The subsequent understanding [8–10] of this scaling fromthe non-linear evolution equations in QCD — the Balitsky-JIMWLK hierarchy [11–17] and its mean fieldapproximation known as the Balitsky-Kovchegov (BK) equation [18] — has greatly increased our confidencein the validity of the pQCD approach to gluon saturation as a valuable tool for phenomenology.

∗Corresponding authorEmail addresses: edmond.iancu@cea.fr (E. Iancu), jose-daniel.madrigal-martinez@cea.fr (J.D. Madrigal),

amh@phys.columbia.edu (A.H. Mueller), gregory.soyez@cea.fr (G. Soyez), trianta@ectstar.eu (D.N. Triantafyllopoulos)

Preprint submitted to Elsevier July 1, 2018

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Over the next years, new ‘dipole models’, of increasing sophistication, have emerged. On one hand, theywere better rooted in perturbative QCD, thus reflecting the overall progress of the theory [19–23]. On theother hand, they were better constrained by the advent of new data at HERA, of higher precision. Finally,they extended the scope of the ‘saturation models’ to other observables, like diffractive structure functionsand particle production in heavy ion collisions. Such extensions required more elaborated versions of thedipole model, including impact parameter dependence [24–27] and heavy quarks [25, 28].

For several years, the theory of high-energy scattering with high gluon density was known only to leadinglogarithmic accuracy in pQCD, which is insufficient for direct applications to phenomenology. For instance,the essential running coupling corrections enter the high energy evolution only at next-to-leading order (seebelow for details). To cope with that, the first generations of dipole models involved phenomenologicalparametrizations for the dipole amplitude, which were rather ad-hoc, albeit sometimes inspired by solutionsto the BK-JIMWLK equations. For instance, the ‘IIM’ fit in [20] attempted to capture some general featuresof the non-linear evolution, like geometric scaling with an anomalous dimension and the BFKL diffusion,that were expected to hold beyond leading order [9, 29]. However, the situation has changed in the recentyears, when the next-to-leading corrections to the BK and JIMWLK equations have progressively becomeavailable [30–35]. This opened the possibility for new fits in which the evolution of the dipole amplitudewith increasing energy is completely fixed by the theory and only the initial condition at low energy stillrequires some modeling involving free parameters. In that respect, the situation of modern ‘dipole fits’becomes comparable in spirit to that of the more traditional fits based on the DGLAP equation.

So far, this strategy has been applied [21–23, 36] only at the level of the “running coupling BK equation”(rcBK) — an improved version of the LO BK equation which resums all-order corrections associated withthe running of coupling, with some scheme dependence though [30–32]. These corrections are numericallylarge, since enhanced by a large transverse (or ‘collinear’) logarithm, and their resummation within the BKequation has important consequences on the evolution — it significantly slows down the growth of the dipoleamplitude with increasing energy [9, 29, 37]. This last feature was indeed essential for the success of theHERA fits based on rcBK [21–23, 36]. The state of the art in that sense is the “AAMQS” fit in [22], whichprovides a good description of the most recent HERA data [38] (the combined analysis by H1 and ZEUSfor the ep reduced cross-section, which is characterized by very small error bars), with a number of freeparameters which varies from 4 to 7 (depending upon whether heavy quarks are included in the fit, or not).

However, the running of the QCD coupling is not the only source of large (but formally higher-order)perturbative corrections to the LO BK, or JIMWLK, equations. Besides the running coupling corrections,the full next-to-leading order (NLO) corrections to the BK equation, as computed in [33], feature othercontributions which are enhanced by potentially large, single or double, transverse logarithms. Such termswere indeed expected, given our experience with the NLO version [39–44] of the BFKL equation [45–47](the linearized version of the BK equation valid when the scattering is weak). The NLO BFKL correctionsare numerically large and thus render the small–x evolution, at LO and NLO, void of any predictive power.There is no reason to expect this problem to be cured, or even alleviated, by the inclusion of the non-linearterms describing unitarity corrections [29, 48]: the collinear logarithms are generated by integrating overregions in phase-space where the dipole size is small and the scattering is weak. This has been indeedconfirmed by the first numerical study of the NLO BK equation [49], which showed that the evolution isunstable (the scattering amplitude decreases with increasing energy and can even turn negative) and thatthe main source for such an instability is the large double-logarithmic correction.

This difficulty reflects the existence of large radiative corrections of higher orders in αs, which formally lieoutside the scope of the high-energy evolution (since generated by the transverse phase-space), but in practicespoil the convergence of the perturbation theory and hence must be kept under control via appropriateresummations. In a previous publication [50], we have devised a resummation scheme which deals with thelargest such corrections — those where each power of αs is accompanied by a double transverse logarithm.Our strategy relies on explicit calculations of Feynman graphs and results in an effective evolution equation— a collinearly improved version of LO BK equation —, in which both the kernel and the initial conditionreceive double-logarithmic corrections to all orders. This scheme differs from the ‘collinear resummations’previously proposed in the context of NLO BFKL [51–55] in that it is explicitly formulated in the transversecoordinate space, rather than in Mellin space, and hence it is consistent with the non-linear structure of the

2

BK equation. Besides, our equation is local in ‘rapidity’ (the logarithm of the energy, which plays the roleof the evolution variable), a property which in this context is rather remarkable since the physics behindthe double collinear logarithms is the time-ordering of subsequent, soft, gluon emissions, which is genuinelynon-local.1 The first numerical studies of this collinearly-improved BK equation demonstrate the essentialrole played by the resummation in both stabilizing and slowing down the evolution [50, 57].

In this paper, we shall provide the first phenomenological test of our resummation scheme, by using it infits to the inclusive HERA data. To that aim, it will be important to first extend this scheme to also includethe single transverse logarithms which appear in the NLO correction to the BK equation — that is, theNLO terms expressing the first correction to the DGLAP splitting kernel beyond the small–x approximationand those associated with the one-loop running of the coupling. Indeed, such single-log effects must be keptunder control to ensure a good convergence of the perturbative expansion. Besides, the inclusion of runningcoupling effects is essential for the description of the data, as well known.

The resummation of the DGLAP logarithms to the order of interest turns out to be rather straight-forward: it amounts to adding an anomalous dimension (a piece of the leading-order DGLAP anomalousdimension) to both the resummed kernel and the resummed initial condition. For the running couplingcorrections, the situation turns out to be more subtle since, strictly speaking, they cannot be encoded intoan equation which is local in rapidity. This being said, and following the standard strategy in the literature,we shall propose various schemes for introducing a running coupling directly in the local evolution equationand test these schemes via fits to the HERA data.

After these additional resummations, we are led to a new, more refined, version for the ‘collinearly im-proved BK equation’, namely Eq. (9) below, which will be our main tool for phenomenology. By construction,this equation resums the double-logarithmic corrections completely — meaning to all orders in αs ≡ αsNc/π(αs is the QCD coupling and Nc is the number of colors) and with the right symmetry factors —, whereasthe single-logarithmic terms are resummed only partially (but in such a way to include the respective termsto NLO). It is rather straightforward to extend our resummed equation to full NLO accuracy, by adding theremaining corrections of O(α2

s), as computed in [33]. But the ensuing equation would be very cumbersometo use in practice, due to the intricate, non-local and non-linear, structure of the pure α2

s corrections. Inthis first analysis, we shall adopt the viewpoint that the most important higher-order contributions (say, inview of phenomenology) are those enhanced by collinear logs, as explicitly resummed in Eq. (9), and thatthe pure α2

s effects are truly small and can be effectively taken care of via the fitting procedure. A similarviewpoint has been advocated in previous fits based on rcBK, but given the importance of the collinearlogarithms, this assumption was not so well motivated and led indeed to some tensions in the respective fits,as we shall later explain.

Using numerical solutions to this collinearly improved BK equation together with suitable forms for theinitial condition, we have performed fits to the HERA data for the ep reduced cross-section [38] at x ≤ 10−2

and Q2 ≤ Q2max, where the upper limit Q2

max on the virtuality Q2 of the exchanged photon is varied between50 GeV2 (a common choice in small–x fits) and 400 GeV2. These fits show several remarkable characteristics.

(i) The fits are indeed successful: for Q2max = 50 GeV2 and two types of initial conditions — GBW–like

[5] and the running-coupling version of the McLerran-Venugopalan (rcMV) model [58] —, we obtain a χ2

per number of data points around 1.2 with only 4 free parameters.(ii) The fits are also very discriminatory: they clearly favor some initial conditions over some others,

and some prescriptions for the running of the coupling over the others. For instance, the standard MV initialcondition, which truly corresponds to a fixed coupling, appears to be disfavored, whereas a more physicalversion of it, including a running coupling, works quite well. The latter works also better than the GBWinitial condition, in the sense that it provides a fit which remains stable up to Q2 = 400 GeV2.

(iii) Our fits alleviate some tensions (in terms of physical interpretation) which were visible in previousfirst based on rcBK [21–23, 36] and could be attributed to the choice to replace all the NLO correctionswith the running of the coupling alone (see also the related discussion in [23]). Notably, our fits preferprescriptions where the QCD coupling αs(µ

2) is running according to the smallest dipole size, they do not

1In fact, a non-local equation to resum the double logarithms has been proposed in [56].

3

require any artificial ‘anomalous dimension’ in the initial condition, and treat the heavy quarks on the samefooting as the light ones, in agreement with general expectations from the dipole factorization.

2. The NLO BK equation and large transverse logarithms

To motivate the resummations that we shall later perform, let us first explicitly exhibit the large trans-verse logarithms which appear when computing the NLO corrections to the BK equation [31–33]. We recallthat the BK equation describes the rapidity evolution of the S-matrix Sxy = 1 − Txy for the scatteringof a color dipole with transverse coordinates (x,y) off a hadronic target. The dipole scattering amplitudeTxy is small in the regime where the target is dilute, but it approaches the unitarity (or ‘black disk’) limitTxy = 1 when the target is dense. The separation between these two regimes is controlled by the saturationmomentum Qs(Y ), which increases with the rapidity difference Y between the projectile and the target.

Neglecting the terms suppressed in the limit of a large number of colors Nc � 1, one finds a closedequation for Sxy, whose NLO version reads as follows [33]

dSxy

dY=αs2π

∫d2z

(x−y)2

(x−z)2(y−z)2

{1 + αs

[b ln(x−y)2µ2 − b (x−z)2 − (y−z)2

(x−y)2ln

(x−z)2

(y−z)2

+67

36− π2

12− 5Nf

18Nc− 1

2ln

(x−z)2

(x−y)2ln

(y−z)2

(x−y)2

]}(SxzSzy − Sxy)

+α2s

8π2

∫d2ud2z

(u−z)4

{− 2 +

(x−u)2(y−z)2 + (x−z)2(y−u)2 − 4(x−y)2(u−z)2

(x−u)2(y−z)2 − (x−z)2(y−u)2ln

(x−u)2(y−z)2

(x−z)2(y−u)2

+(x−y)2(u−z)2

(x−u)2(y−z)2

[1 +

(x−y)2(u−z)2

(x−u)2(y−z)2 − (x−z)2(y−u)2

]ln

(x−u)2(y−z)2

(x−z)2(y−u)2

}(SxuSuzSzy − SxuSuy)

+α2s

8π2

Nf

Nc

∫d2ud2z

(u−z)4

[2− (x−u)2(y−z)2 + (x−z)2(y−u)2 − (x−y)2(u−z)2

(x−u)2(y−z)2 − (x−z)2(y−u)2ln

(x−u)2(y−z)2

(x−z)2(y−u)2

](SxzSuy − SxuSuy) , (1)

where Nf is the number of flavors, b = (11Nc − 2Nf)/12Nc, and αs = αsNc/π, with the QCD coupling αsevaluated at the renormalization scale µ.

There are two main changes in the structure of the evolution equation as we go from LO to NLO.First, the term with a single integration (SI) over the transverse coordinate z only receives a correction oforder O(α2

s) to the kernel, which in particular contains the running coupling corrections proportional to b.Second, there are new terms, of order O(α2

s), which involve a double integration (DI) over the transversecoordinates u and z and which refer to partonic fluctuations involving two additional partons (besides theoriginal quark and antiquark) at the time of scattering. The first such a term, which is independent ofNf , represents fluctuations where both daughter partons are gluons. The S-matrix structure therein, thatis, SxuSuzSzy − SxuSuy, corresponds to the following sequence of emissions: the original dipole (x,y)emits a gluon at u, thus effectively splitting into two dipoles (x,u) and (u,y); then, the dipole (u,y) emitsa gluon at z, thus giving rise to the dipoles (u, z) and (z,y). The ‘real’ term SxuSuzSzy describes thesituation where both daughter gluons interact with the target. The ‘virtual’ term −SxuSuy describes thecase where the gluon at z has been emitted and reabsorbed either before, or after, the scattering. Thisnegative ‘virtual’ term subtracts the equal-point contribution (z = u) from the ‘real’ piece, ensuring thatthe potential ‘ultraviolet’ singularity associated with the factor 1/(u−z)4 in the kernel is truly harmless. Asimilar discussion applies to the second DI term, proportional to Nf , except for the fact that the additionalpartons at the time of scattering are a quark and an antiquark.

In principle, one should be able to undertake the task of solving the NLO BK equation. The hopewould be that the solution would only add a relatively small correction to the LO result. However, this isnot the case since there are terms in the kernels of the NLO equation which can become large in certainkinematic regimes and thus invalidate the strict αs-expansion. One obvious class of such terms contains the

4

corrections proportional to b in the SI term in Eq. (1), which by themselves bring no serious difficulties: aswell known, these corrections can be absorbed into a redefinition of the scale for the running of the coupling,which thus becomes a dynamical scale (see Sect. 4 below for details). Here, we would like to focus on thecorrections enhanced by ‘collinear logarithms’, that is, logarithms associated with the large separation intransverse sizes (or momenta) between successive emissions. These corrections become large only in theweak-scattering regime where all the dipoles are small compared to the saturation scale 1/Qs(Y ) and theequation can be linearized w.r.t. to the (small) scattering amplitude T . This in particular means that onecan ignore the last term, proportional to Nf/Nc, in Eq. (1) since this term vanishes after linearization, asone can easily check (by also using the symmetry of the kernel under the interchange u↔ z).

To be more precise, let us consider the strongly ordered regime

1/Qs � |z − x| ' |z − y| ' |z − u| � |u− x| ' |u− y| � |x− y|, (2)

that is, the parent dipole is the smallest one, a gluon is emitted far away at u, a second one even further atz, but with all possible dipole sizes remaining smaller than the inverse saturation momentum. Wheneverappropriate, we will denote by r, u and z the size of the parent dipole, the size of the dipoles involving uand the size of the dipoles involving z, respectively, with r2 � u2 � z2. By inspection of the SI piece in theNLO BK equation, it is quite obvious that the dominant NLO term is the one involving a double transverselogarithm (DTL), that is, the last term within the square brackets. Still within this regime (2), we canapproximate the scattering matrices in the SI term as follows: SxzSzy−Sxy ' −Txz−Tzy +Txy ' −2T (z),where the second approximate equality follows since the dipole amplitude for a small dipole is roughlyproportional to the dipole size squared. Notice that the net result in the approximation of interest fullycomes from the ‘real’ term, which involves the large daughter dipoles.

What is not immediately obvious is the presence of a single transverse logarithm (STL) coming from theDI term. Let us isolate here the relevant part of the kernel,

MSTL ≡1

8(u−z)4

[−2+

(x−u)2(y−z)2 + (x−z)2(y−u)2 − 4(x−y)2(u−z)2

(x−u)2(y−z)2 − (x−z)2(y−u)2ln

(x−u)2(y−z)2

(x−z)2(y−u)2

]. (3)

To implement the limit in Eq. (2) we can successively write the expression in Eq. (3) as

MSTL '1

8z4

[−2+

2u2 − 2ur cosφ− 3r2

r2 − 2ur cosφln

(1 +

r2 − 2ur cosφ

u2

)]' −6− cos2 φ

12

r2

u2z4→ −11

24

r2

u2z4, (4)

with φ the angle between r and any of the two dipoles involving u. To obtain (4), we have first set all dipolesizes which include z equal to each other, since any subleading term would be suppressed by inverse powersof z. Then the only z dependence left is the one explicit in the prefactor. We have subsequently takenthe limit r � u (by expanding the logarithm to cubic order) and we have finally averaged over the angleφ between the parent dipole and those involving u. Notice that the would-be leading term, of order 1/z4,has cancelled out in these manipulations. The first non-vanishing term, as visible in the r.h.s. of Eq. (4),is suppressed by r2/u2, thus creating the conditions for a logarithmic integration over u. To explicitly seethis, recall that we consider the weak-scattering regime, where the product of S-matrices multiplyingMSTL

can be linearized. This allows us write SxuSuzSzy −SxuSuy ' −Tuz −Tzy +Tuy ' −2T (z). (Once again,the dominant contribution has been generated by the ‘real’ term.) We see that the net scattering amplitudein this approximation is independent of the intermediate dipole size u. Accordingly, when integrating overu, within the range limited by r and z, we find a STL, as anticipated. After also including the LO term andthe NLO one enhanced by the DTL, one finds that the NLO BK equation in the strongly ordered region (2)reduces to

dT (r)

dY= αs

∫ 1/Q2s

r2dz2 r

2

z4

(1− 1

2αs ln2 z

2

r2− 11

12αs ln

z2

r2

)T (z). (5)

It is now clear that, if the daughter dipoles are allowed to become sufficiently large, the NLO contributionsenhanced by large transverse logarithms become comparable to, or larger than, the LO one. In that case,

5

the present perturbative expansion cannot be trusted anymore. To be more explicit, consider a single step∆Y in the evolution with the following, simple but physically meaningful, initial condition

T (r) =

{r2Q2

s for r2Q2s � 1

1 for r2Q2s � 1.

(6)

The z-integration in Eq. (5) becomes logarithmic and gives

∆T (r) = αs∆Y r2Q2

s ln1

r2Q2s

(1− 1

6αs ln2 1

r2Q2s

− 11

24αs ln

1

r2Q2s

). (7)

This shows that, for sufficiently small rQs, such that αs ln2(1/r2Q2s) & 1, the NLO correction becomes larger

than the LO term and the perturbation series is unreliable. In particular, the NLO correction is negativerendering the solution unstable, as indeed observed in numerical solutions [48–50].

3. Resumming the large collinear logarithms

The large NLO corrections that we have singled out in the previous section are the lowest-order examplesof collinearly-enhanced radiative corrections, which occur to all orders and spoil the convergence of theperturbation theory. When the separation between the transverse scales of the projectile and the targetis large enough, as is actually the case in the DIS kinematics at HERA, the higher-order terms of thistype become more important than the pure α2

s NLO terms (i.e. the contributions of O(α2s) which are not

amplified by any transverse logarithms). From now on, we shall focus on this situation, discarding the pureα2s NLO corrections, but focusing on the resummation of the large transverse logarithms to all orders. In

this section, we consider both the single and double collinear logarithms, thus following and expanding ourrecent results in [50]. In the next section, we shall explain how the running coupling corrections can beincluded in this scheme.

In Ref. [50] we have devised a strategy for resumming double-logarithmic corrections to the BK equationto all orders. Our main observation was that these corrections are generated by the diagrams commonto the BFKL and DGLAP evolutions — i.e. the Feynman graphs of light-cone perturbation theory inwhich the successive gluon emissions are strongly ordered in both longitudinal momenta and transversemomenta (or ‘dipole sizes’) — after enforcing the additional constraint that the emissions must also beordered in lifetimes (or, equivalently, in light-cone energies [56]). Concerning the single collinear logarithms,it is intuitively clear that they must represent DGLAP-like corrections to BFKL, ‘small-x’, emissions. Forinstance, the effect of order α2

s∆Y ρ2 visible in Eq. (7) is the result of a sequence of two emissions: one

small-x emission (in the double logarithmic regime) yielding a contribution ∝ αs∆Y ρ, and a DGLAP-likeemission, characterized by strong ordering in dipole sizes (see eq. Eq. (2)) and which gives an effect oforder αsρ. Here, ρ ≡ ln(1/Q2

sr2) is the large logarithm generated by the transverse phase-space. This

scenario is corroborated by the following observation: the numerical coefficient A1 ≡ 11/12 in front of theSTL in Eq. (5) can be recognized as the second-order term in the small ω expansion of the relevant linearcombination of DGLAP anomalous dimensions:

PT(ω) =

∫ 1

0

dz zω[Pgg(z) +

CF

NcPqg(z)

]=

1

ω−A1 +O

(ω,

Nf

N3c

)with A1 =

11

12. (8)

Recalling that one needs one factor of 1/ω in order to generate a small-x logarithm ∆Y = ln(1/x), one seesthat the NLO effect ∼ α2

s∆Y ρ2 is indeed produced by combining the singular (1/ω) piece of one emission

with the first non-singular piece (A1) of another one. This discussion also instructs us about the strategyto follow in order to resum such STLs to all orders: it suffices to include this piece A1 as an anomalousdimension, i.e. as an extra power-law suppression, in the evolution kernel previously obtained in Ref. [50].

6

α���α ���

��

� � � � �

���

���

���

���

|�-�|

|�-�|=�ϕ=�

(a)

α���α ���

��

� � � � �

���

���

���

���

|�-�|

|�-�|=�ϕ=π /�

(b)

α���α ���

��

� π /� π /� �π /� π

���

���

���

���

ϕ

|�-�|=�|�-�|=���

(c)

Figure 1: Running coupling for various schemes and configurations. (a) As a function of the daughter dipolesize |x − z|, with φ = 0 the angle between the parent dipole x − y and the daughter one x − z. (b) Thesame with φ = π/6. (c) As a function of the angle φ for fixed daughter dipole size |x − z| = 1.5. Black(continuous): The minimal dipole scheme as defined in Eq. (11). Red (dashed): The “fac” scheme as givenin Eq. (12). Blue (dotted): The Balitsky scheme [32]. In all cases the parent dipole size is |x− y| = 1, thecoupling is smoothly frozen at the value 0.7 and ΛQCD = 0.2.

We are thus led to the following, collinearly-improved, version of the BK equation,

dTxydY

=αs2π

∫d2z

(x−y)2

(x−z)2(z−y)2

[(x−y)2

min{(x−z)2, (y−z)2}

]±αsA1

KDLA

(√LxzrLyzr

)×(Txz + Tzy − Txy − TxzTzy

), (9)

where the overall kernel is written as a product of three factors: the familiar dipole kernel which appearsalready at leading order, the ‘DLA kernel’, resuming the double collinear logs to all orders [50]

KDLA(ρ) =J1

(2√αsρ2

)√αsρ2

= 1− αsρ2

2+

(αsρ2)2

12+ · · · , (10)

evaluated at ρ =√LxzrLyzr, with Lxzr ≡ ln[(x− z)2/r2], and a new factor, which features the exponent

±αsA1 (the positive sign in the exponent is taken when |x−y| < min{|x−z|, |y−z|} and the negative signotherwise), which expresses the contribution of the single collinear logarithms.

From the above discussion, is should also be clear that the present resummation of STLs is only partial:it refers to the particular class of such corrections which are generated by the first non-singular piece in theexpansion in Eq. (8). The higher terms in this ω–expansion will produce single collinear logarithms too, butonly starting at higher orders in perturbation theory (NNLO or higher). At the level of the BFKL equation,more complete resummations of the single logarithms have been devised in [52–54], but so far it is not clearhow to extend these resummation schemes to a non-linear evolution equation like BK.

Returning to Eq. (9), the tilde symbol in Txy is intended to remind that this is truly a suitable analyticcontinuation of the dipole amplitude which coincides with the physical quantity Txy only for ρ < Y . Forρ > Y , the physical amplitude can be obtained by either solving an equation non-local in Y , or by matchingonto the solution to the DGLAP equation [50]. However, explicit numerical studies at DLA level have shownthat the solution Txy to Eq. (9) remains very close to the actual physical amplitude, including for ρ > Y . Forthis reason, we shall ignore this subtlety (and the related issue of the resummation in the initial condition)for the purpose of the fits to be constructed in the next section. We shall return to a more detailed studyof these issues in a forthcoming publication [59].

4. Prescriptions for the running of the coupling

The last source of potentially large NLO corrections to the BK equation are the running couplingcorrections, i.e. the logarithmic terms proportional to b in the SI term in Eq. (1). Such terms can grow large

7

when the scales in their arguments are very disparate. More precisely, the first logarithm can be problematicwhen r is much smaller or much larger than 1/µ, while the second when the soft gluon at z is collinear tothe quark or the antiquark composing the parent dipole. We need to choose µ in such a way to cancel thesepotentially large logarithms, which could otherwise spoil the convergence of the perturbative expansion2. Itis clear that there is not a unique choice, but in QCD one usually expects the hardest scale to determinethe running of the coupling. Indeed, a quick inspection shows that the smallest dipole prescription

αmin = αs(rmin) with rmin = min{|x−y|, |x−z|, |y−z|} (11)

cancels the large logarithms in all kinematic regions.Another possibility is to choose µ so that all the terms with coefficient α2

s b vanish. Given that in thecurrent work we neglect all finite (i.e. not enhanced by a large logarithm) α2

s terms, this looks like what iscalled the “fastest apparent convergence” (fac) scheme [60–62]. It is convenient in the sense that one is leftwith just the leading term in αs. We find that

αfac =

[1

αs(|x−y|)+

(x−z)2 − (y−z)2

(x−y)2

αs(|x−z|)− αs(|y−z|)αs(|x−z|)αs(|y−z|)

]−1

, (12)

and it is an easy exercise to show that it reduces to the minimal dipole choice αs(rmin) in all limits whereone of the three dipoles is much smaller than the other two.

In this work, we shall use both above schemes. Let us add that the most popular prescription, widelyused so far in phenomenological applications, is the one due to Balitsky [32], and reads

αBal = αs(|x−y|)[1 +

αs(|x−z|)− αs(|y−z|)αs(|x−z|)αs(|y−z|)

αs(|x−z|)(y−z)2 − αs(|y−z|)(x−z)2

(x−y)2

], (13)

but it will not be adopted here for a number of reasons. First, it is based on an extrapolation to all ordersof a coordinate space kernel which includes the α2

s b terms above as well as α3s b

2 corrections. At this order,we would also expect corrections proportional to the two-loop beta function. Second, even though it alsoreduces to αs(rmin) in the extreme kinematical limits, it does that very slowly for large daughter dipoles (incertain configurations) and this leads to an unphysically small coupling in a large region of phase space, ascan be seen in the respective plots in Fig. 1. Finally, and perhaps as a result of the above drawbacks, whenused in fitting the DIS data, it gives a much worse fit than Eqs. (11) and (12) and with fit parameters whichtake somewhat unnatural values.

5. Fits to the HERA data

We now turn to the description of the HERA reduced cross-section measurements using the resummedBK equation. To this aim several ingredients first have to be specified.

Initial condition. We must fix the initial condition for the dipole amplitude at some Y0, which afterwardswill be evolved towards higher rapidities using Eq. (9). We consider two choices: the simple parametrisationof the Golec-Biernat and Wusthoff (GBW) [5] type

T (Y0, r) =

{1− exp

[−(r2Q2

0

4

)p]}1/p

(14)

and the running-coupling version of the McLerran-Venugopalan (rcMV) model [58]

T (Y0, r) =

{1− exp

[−(r2Q2

0

4αs(r)

[1 + ln

(αsat

αs(r)

)])p]}1/p

. (15)

2It is rather important to point out here that µ should cancel only these logarithms and not those discussed earlier whichare of different physical origin. Of course one can proceed to such a choice and cancel all the NLO logarithms, but the resultwill be extremely unstable w.r.t. small variations of µ.

8

init RC sing. χ2 per data point parameterscdt. schm logs σred σccred FL Rp[fm] Q0[GeV] Cα pGBW small yes 1.135 0.552 0.596 0.699 0.428 2.358 2.802GBW fac yes 1.262 0.626 0.602 0.671 0.460 0.479 1.148rcMV small yes 1.126 0.565 0.592 0.707 0.633 2.586 0.807rcMV fac yes 1.228 0.647 0.594 0.677 0.621 0.504 0.541GBW small no 1.121 0.597 0.597 0.716 0.414 6.428 4.000GBW fac no 1.164 0.609 0.594 0.697 0.429 1.195 4.000rcMV small no 1.093 0.539 0.594 0.718 0.647 7.012 1.061rcMV fac no 1.132 0.550 0.591 0.699 0.604 1.295 0.820

Table 1: χ2 and values of the fitted parameters entering the description of the HERA data. The fit includesthe 252 σred data points. The quoted χ2 for σccred and FL are obtained a posteriori.

It is worth noticing that, as dictated by collinear physics, there is no anomalous dimension in the aboveinitial conditions. The extra parameter p determines the shape of the amplitude close to saturation and itsapproach towards unitarity.

Running coupling. We consider the two prescriptions given by Eqs. (11) and (12). For the explicit expressionof the strong coupling in coordinate space in terms of r we introduce a fudge factor as in [22], namely

αs(r) =1

bNfln[4C2

α/(r2Λ2

Nf)] , (16)

with bNf= (11Nc − 2Nf)/12π. This fudge factor is also included in the rcMV type initial condition in (15).

The Nf -dependent Landau pole is obtained by imposing αs(M2Z) = 0.1185 at the scale of the Z mass [63]

and continuity of αs at the flavour thresholds, using mc = 1.3 GeV and mb = 4.5 GeV. To regularise theinfrared behaviour, we have decided to freeze αs at a value αsat = 1 and we have checked explicitly thatreducing this down to, for example, 0.7 does not affect the fit in any significant manner.

Note that we do not include any form of resummation or matching for ln 1/r2 > Y , as introduced in[50], in these initial conditions. One of the reasons for not doing so is that the extra factor in the initialcondition can always be reabsorbed in a re-parametrisation. Furthermore, a proper matching at small dipolesizes, suited for phenomenological studies, would require a careful treatment of the small-dipole region. Inthat respect, the resummed BK evolution is expected to perform a better job than a fixed matching witha fixed asymptotic behaviour. We leave a better treatment, e.g. a genuine matching to DGLAP evolution,for future work.

Rapidity evolution. Of course this is determined by the resummed BK equation given in (9). Here, we againconsider two separate cases, one in which the evolution resums only the leading double logarithms and onein which it also includes the single ones.

From the dipole amplitude to observables. Once we have the dipole amplitude for all rapidities and dipolesizes, we use the standard dipole formalism to obtain the physical observables:

σγ∗p

L,T(Q2, x) = 2πR2p

∑f

∫d2r

∫ 1

0

dz∣∣Ψ(f)

L,T(r, z;Q2)∣∣2 T (ln 1/xf , r), (17)

where the transverse and longitudinal virtual photon wavefunctions read∣∣Ψ(f)L (r, z;Q2)

∣∣2 = e2q

αemNc2π2

4Q2z2(1− z)2K20 (rQf ), (18)∣∣Ψ(f)

T (r, z;Q2)∣∣2 = e2

q

αemNc2π2

{[z2 + (1− z)2

]Q2fK

21 (rQf ) +m2

fK20 (rQf )

}. (19)

9

init RC sing. χ2/npts for Q2max

cdt. schm logs 50 100 200 400GBW small yes 1.135 1.172 1.355 1.537GBW fac yes 1.262 1.360 1.654 1.899rcMV small yes 1.126 1.170 1.182 1.197rcMV fac yes 1.228 1.304 1.377 1.421GBW small no 1.121 1.131 1.317 1.487GBW fac no 1.164 1.203 1.421 1.622rcMV small no 1.093 1.116 1.106 1.109rcMV fac no 1.131 1.181 1.171 1.171

Table 2: Evolution of the fit quality when including data at larger Q2 (in GeV2).

In the above we have introduced the customary notation Q2f = z(1− z)Q2 +m2

f , xf = x(1 + 4m2f/Q

2), andwe have assumed a uniform distribution over a disk of radius Rp in impact parameter space. The sum in (17)runs over all quark flavours and we will include the contributions from light quarks with mu,d,s = 100 MeVas well as from the charm quark with mc = 1.3 GeV. From the longitudinal and transverse cross-sections,we can deduce the reduced cross-section and the longitudinal structure function as

σred =Q2

4π2αem

[σγ

∗pT +

2(1− y)

1 + (1− y)2σγ

∗pL

], (20)

FL =Q2

4π2αemσγ

∗pL . (21)

When the quark masses, the value of the strong coupling at the Z mass and its frozen value in theinfrared have been fixed, we are left with 4 free parameters according to our choice of initial condition: Rpthe “proton radius”, Q0 the scale separating the dilute and dense regimes, Cα the fudge factor in the runningcoupling in coordinate space, and p which controls the approach to saturation in the initial condition.

We have fitted these parameters to the combined HERA measurements of the reduced photon-protoncross-section [38]. Since the BK equation is applicable only at small-x, we have limited ourselves to theregion x ≤ 0.01. We note that since Eq. (17) probes dipoles at the rapidity ln 1/xf , the exact cut we imposeis xc ≤ 0.01 since the most constraining cut comes from the charm, the most massive quark we include inour model. Accordingly, our initial condition for the BK evolution corresponds to x = 0.01. Furthermore,since we do not expect the BK equation to capture the full collinear physics, we impose the upper boundQ2 < Q2

max. By default we will use Q2max = 50 GeV2 but we will also give results for extensions to larger

Q2. In the default case we have a total of 252 points included in the fit. We have added the statistical andsystematic uncertainties in quadrature.3

The results of our fits for the 23 = 8 cases, depending on the initial condition, the running couplingprescription and the inclusion or not of single logarithms in the kernel, are presented in Table. 1. The tableincludes the parameter values obtained from fitting the σred data and, besides the fit χ2, it also indicatesthe χ2 obtained a posteriori for the latest σccred [64] and FL [65] measurements. These results deserve a fewimportant comments.

(i) In general, the overall quality of the fit is very good, reaching χ2 per point around 1.1-1.2.

(ii) Apart from a few small exceptions (see below), all the parameters take acceptable values of order one.Note that we have manually bounded p between 0.25 and 4. Whenever it reached the upper limit,larger values only led to minor improvements in the quality of the fit.

3A more involved treatment of the correlated systematic uncertainties leads to similar results with slightly worse χ2 perpoints (about 0.04).

10

the

ory

/da

ta

0.9

0.95

1

1.05

1.1 0.045,0.065 0.085 0.1 0.15 0.2

the

ory

/da

ta

x

0.9

0.95

1

1.05

1.1

10-6

10-5

10-4

0.25

x10

-610

-510

-4

0.35

x10

-610

-510

-4

0.4

x10

-610

-510

-4

0.5

x10

-610

-510

-4

0.65

the

ory

/da

ta

0.9

0.95

1

1.05

1.1 0.85 1.2 1.5 2 2.7

the

ory

/da

ta

x

0.9

0.95

1

1.05

1.1

10-5

10-4

10-3

3.5

x10

-510

-410

-3

4.5

x10

-510

-410

-3

6.5

x10

-510

-410

-3

8.5

x10

-510

-410

-3

10

the

ory

/da

ta

0.9

0.95

1

1.05

1.112 15 18 22 27

the

ory

/da

ta

x

0.940.960.98

11.021.041.06

10-3

10-2

35

x10

-310

-2

45 χ2

1.131.231.091.13

λs

0.200.240.180.20

HERA datasmall, DL+SLfac, DL+SLsmall, DLfac, DL

Figure 2: Description of the HERA data obtained by the fits using the rcMV initial condition. Each boxcorresponds to a given value of Q2 as indicated (in GeV2) in the top-right corner. For each fit we plotthe ratio of the prediction to the central experimental value. The (green) band represents the experimentaluncertainty.

11

(iii) The two initial conditions give similar results, with a slight advantage for the rcMV option, which islikely due to the extra parameter. Note that for a standard MV-type of initial condition T (Y0, r) ={1− exp[−(r2Q2

0/4 [c+ ln(1 + 1/rΛ)])p]}1/p, we have not been able to obtain a χ2 per point below 1.3and the parameters, typically c or p, tend to take unnatural values.

(iv) As far as the running-coupling prescription is concerned, the smallest dipole prescription given inEq. (11) tends to give somewhat better fits than the “fac” prescription given in Eq. (12). This can beseen as an estimate of subleading corrections (including the pure α2

s NLO terms) that we neglect inthe present fit. Note also that we have not been able to reach a fit of equivalent quality and robustnesswith the Balitsky prescription.

(v) The resummation of the single logarithms tends to yield slightly larger values for χ2, but the differenceis too small to be significative (at least, without performing a full NLO analysis). Perhaps moresignificantly, this resummation leads to more physical values for some of the parameters, especially Cαfor the smallest dipole prescription and p for the GBW initial condition. These findings are consistentwith the expectation that, once properly resummed, single logarithms should have only a modestimpact. Recall however that their resummation is a crucial step towards a full NLO fit — failing todo so could lead to instabilities similar to those observed when double logarithms are not resummed.

(vi) The fit remains stable when varying the parameters we have imposed by hand. For example, usingαsat = 0.7 instead of 1 has no significant effect on the fit. Varying the light quark masses within therather wide range 0 ≤ mu,d,s ≤ 140 MeV only slightly changes the quality of the fit. For instance,taking one of our best fits (rcMV initial condition, the smallest dipole prescription for the running of thecoupling, and resummation of the single logarithms), we have found χ2 = {1.180, 1.153, 1.126, 1.159}when choosing mu,d,s = {0, 50, 100, 140} MeV, respectively. This lack of sensitivity to the light quarkmasses is likely a consequence of saturation, which reduces the dependence of the DIS cross-section tovery large dipole fluctuations. (The corresponding amplitudes reach the unitarity, or ‘black disk’, limitT = 1, so they are independent of the size r of the dipoles fluctuations, as regulated at low Q2 by thequark masses; see also the discussion of Fig. 3 below.) Also, we have obtained an equally good fit withthe slightly larger value mc = 1.4 GeV for the mass of the charm quark, although the quality starteddeteriorating for significantly larger values mc ≥ 1.6 GeV. Very similar findings have been reportedfor the saturation fits in [26].

(vii) Trying to extend the fit to larger Q2 shows an interesting behaviour as seen from Table 2. While theχ2 obtained using the GBW initial condition increase when including higher-Q2 data, the fits usingthe rcMV initial condition remain stable. We suspect that this is due to the fact that this choice ofinitial condition stays closer to the expected physics at high Q2.

In Fig. 2 one can see the quality of our fit and the extracted values of the evolution parameter λs =d lnQ2

s/dY . In Fig. 3 we show the value of the saturation momentum in the (x,Q2)-plane on top of thedata points as well as a few selected initial conditions for the fit. Note that amplitudes which a priorihave different functional forms, cf. Eqs. (14) and (15), look nevertheless quite similar in shape (at least indouble-logarithmic scale) when plotted for the particular values of the parameters that are selected by thefits.

To conclude, this work can be seen as the first description of small-x DIS data which includes mandatoryperturbative QCD ingredients in that region: leading-order small-x evolution, the resummation of largetransverse logarithms, and saturation corrections4. The dipole amplitude obtained from our fits to inclusiveDIS can in principle be used to compute several other observables, like particle multiplicity in hadroniccollisions, the diffractive structure functions, the elastic production of vector mesons, or the forward particleproduction in heavy-ion collisions. This is certainly not the end of the story: beyond what we have includedhere, it would be interesting to add the pure α2

s NLO corrections to the BK evolution kernel, thus obtaininga genuine resummed NLO-BK fit, and to perform a proper matching between this small-x evolution and aDGLAP-like evolution at large Q2 and large x. These steps go beyond the scope of the present paper andare left for future studies.

4It would be an interesting exercise to see what happens if one switches off the non-linear corrections in Eq. (9). Given ourasymmetric choice of frame, justified by saturation physics, this may however require extra work. See also [66].

12

10-1

100

101

10-6

10-5

10-4

10-3

10-2

Q2 o

r 4

/r2 [

Ge

V2]

x

dataGBW,smallGBW,facrcMV,smallrcMV,fac

10-5

10-4

10-3

10-2

10-1

100

10-2

10-1

100

101

DL+SL resummed

T(r

,Y=

0)

r [GeV-1

]

GBW,smallGBW,facrcMV,smallrcMV,fac

Figure 3: Left: value of the saturation momentum, defined for each rapidity as 2/rs(Y ) with T (rs(Y ), Y ) =1/2. For comparison, we have overlaid the experimental data points from the HERA dataset. Right: plotof the corresponding initial conditions for the rapidity evolution.

Acknowledgements

This work is supported by the European Research Council under the Advanced Investigator Grant ERC-AD-267258 and by the Agence Nationale de la Recherche project # 11-BS04-015-01. The work of A.H.M. issupported in part by the U.S. Department of Energy Grant # DE-FG02-92ER40699. G.S. wishes to thankJavier Albacete and Guilherme Milhano for helpful discussions on the AAMQS results.

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